reserve c, c1, c2, d, d1, d2, e, y for Real,
  k, n, m, N, n1, N0, N1, N2, N3, M for Element of NAT,
  x for set;

theorem :: (Part 5, O(2^2n) c O(n!))
  ex s being eventually-positive Real_Sequence st s = seq_a^(2,2,0) &
  Big_Oh(s) c= Big_Oh(seq_n!(0)) & not Big_Oh(s) = Big_Oh(seq_n!(0))
proof
  reconsider f = seq_a^(2,2,0)as eventually-positive Real_Sequence;
  set g = seq_n!(0);
  take f;
  thus f = seq_a^(2,2,0);
  set h = f/"g;
A1: now
    let p be Real;
    assume
A2: p > 0;
    set N = max(10,[/9+log(2,1/p)\]);
A3: N >= 10 by XXREAL_0:25;
A4: N is Integer by XXREAL_0:16;
A5: N >= [/9+log(2,1/p)\] by XXREAL_0:25;
    N in NAT by A3,A4,INT_1:3;
    then reconsider N as Nat;
    take N;
    let n be Nat;
A6:  n in NAT by ORDINAL1:def 12;
A7: [/9+log(2,1/p)\] >= 9+log(2,1/p) by INT_1:def 7;
    assume
A8: n >= N;
    then n >= [/9+log(2,1/p)\] by A5,XXREAL_0:2;
    then n >= 9+log(2,1/p) by A7,XXREAL_0:2;
    then n-9 >= log(2,1/p) by XREAL_1:19;
    then 2 to_power (n-9) >= 2 to_power log(2,1/p) by PRE_FF:8;
    then 2 to_power (n-9) >= 1/p by A2,POWER:def 3;
    then
A9: 1 / (2 to_power (n-9)) <= 1/(1/p) by A2,XREAL_1:85;
A10: h.n = f.n/g.n by Lm4
      .= (2 to_power (2*n+0)) / g.n by Def1,A6
      .= (2 to_power (2*n+0)) / ((n+0)!) by Def4
      .= (2 to_power (2*n)) / (n!);
    n >= 10 by A3,A8,XXREAL_0:2;
    then h.n < 1 / (2 to_power (n-9)) by A10,Lm34,A6;
    then h.n < p by A9,XXREAL_0:2;
    hence |.h.n-0.| < p by A10,ABSVALUE:def 1;
  end;
  then
A11: h is convergent by SEQ_2:def 6;
  then
A12: lim h = 0 by A1,SEQ_2:def 7;
  then not g in Big_Oh(f) by A11,ASYMPT_0:16;
  then
A13: not f in Big_Omega(g) by ASYMPT_0:19;
  f in Big_Oh(g) by A11,A12,ASYMPT_0:16;
  hence thesis by A13,Th4;
end;
